CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting Fractions 41 2.3.3 Converting between Power-of-Two Radices 44 2.4 Signed Integer Representation 44 2.4.1 Signed Magnitude 44 2.4.2 Complement Systems 49 2.5 Floating-Point Representation 55 2.5.1 A Simple Model 56 2.5.2 Floating-Point Arithmetic 58 2.5.3 Floating-Point Errors 59 2.5.4 The IEEE-754 Floating-Point Standard 61 2.6 Character Codes 62 2.6.1 Binary-Coded Decimal 62 2.6.2 EBCDIC 63 2.6.3 ASCII 63 2.6.4 Unicode 65 2.7 Codes for Data Recording and Transmission 67 2.7.1 Non-Return-to-Zero Code 68 2.7.2 Non-Return-to-Zero-Invert Encoding 69 2.7.3 Phase Modulation (Manchester Coding) 70 2.7.4 Frequency Modulation 70 2.7.5 Run-Length-Limited Code 71 2.8 Error Detection and Correction 73 2.8.1 Cyclic Redundancy Check 73 2.8.2 Hamming Codes 77 2.8.3 Reed-Soloman 82 Chapter Summary 83 CMPS375 Class Notes Page 1/ 16 by Kuo-pao Yang 2.1 Introduction 37 • This chapter describes the various ways in which computers can store and manipulate numbers and characters. • Bit: The most basic unit of information in a digital computer is called a bit, which is a contraction of binary digit. • Byte: In 1964, the designers of the IBM System/360 main frame computer established a convention of using groups of 8 bits as the basic unit of addressable computer storage. They called this collection of 8 bits a byte. • Word: Computer words consist of two or more adjacent bytes that are sometimes addressed and almost always are manipulated collectively. Words can be 16 bits, 32 bits, 64 bits. • Nibbles: Eight-bit bytes can be divided into two 4-bit halves call nibbles. 2.2 Positional Numbering Systems 38 • Radix (or Base): The general idea behind positional numbering systems is that a numeric value is represented through increasing powers of a radix (or base). System Radix Allowable Digits --------------------------------------------------------------------- Decimal 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Binary 2 0, 1 Octal 8 0, 1, 2, 3, 4, 5, 6, 7 Hexadecimal 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F FIGURE 2.1 Some Number to Remember EXAMPLE 2.1 Three numbers represented as powers of a radix. 2 1 0 -1 -2 243.5110 = 2 * 10 + 4 * 10 + 3 * 10 + 5 * 10 + 1 * 10 2 1 0 2123 = 2 * 3 + 1 * 3 + 2 * 3 = 2310 4 3 2 1 0 101102 = 1 * 2 + 0 * 2 + 1 * 2 + 1 * 2 + 0 * 2 = 2210 CMPS375 Class Notes Page 2/ 16 by Kuo-pao Yang 2.3 Decimal to Binary Conversions 38 • There are two important groups of number base conversions: 1. Conversion of decimal numbers to base-r numbers 2. Conversion of base-r numbers to decimal numbers 2.3.1 Converting Unsigned Whole Numbers 39 • EXAMPLE 2.3 Convert 10410 to base 3 using the division-remainder method. 10410 = 102123 • EXAMPLE 2.4 Convert 14710 to binary 14710 = 100100112 • A binary number with N bits can represent unsigned integer from 0 to 2n – 1. • Overflow: the result of an arithmetic operation is outside the range of allowable precision for the give number of bits. 2.3.2 Converting Fractions 41 • EXAMPLE 2.6 Convert 0.430410 to base 5. 0.430410 = 0.20345 • EXAMPLE 2.7 Convert 0.3437510 to binary with 4 bits to the right of the binary point. Reading from top to bottom, 0.3437510 = 0.01012 to four binary places. We simply discard (or truncate) our answer when the desired accuracy has been achieved. • EXAMPLE 2.8 Convert 31214 to base 3 First, convert to decimal 312144 = 21710 Then convert to base 3 We have 31214 = 220013 2.3.3 Converting between Power-of-Two Radices 44 • EXAMPLE 2.9 Convert 1100100111012 to octal and hexadecimal. 1100100111012 = 62358 Separate into groups of 3 for octal conversion 1100100111012 = C9D16 Separate into groups of 4 for octal conversion CMPS375 Class Notes Page 3/ 16 by Kuo-pao Yang 2.4 Signed Integer Representation 44 • By convention, a “1” in the high-order bit indicate a negative number. 2.4.1 Signed Magnitude 44 • A signed-magnitude number has a sign as its left-most bit (also referred to as the high-order bit or the most significant bit) while the remaining bits represent the magnitude (or absolute value) of the numeric value. • N bits can represent –(2n-1 - 1) to 2n-1 -1 • EXAMPLE 2.10 Add 010011112 to 001000112 using signed-magnitude arithmetic. 010011112 (79) + 001000112 (35) = 011100102 (114) There is no overflow in this example • EXAMPLE 2.11 Add 010011112 to 011000112 using signed-magnitude arithmetic. An overflow condition and the carry is discarded, resulting in an incorrect sum. We obtain the erroneous result of 010011112 (79) + 011000112 (99) = 01100102 (50) • EXAMPLE 2.12 Subtract 010011112 from 011000112 using signed-magnitude arithmetic. We find 0110000112 (99) - 010011112 (79) = 000101002 (20) in signed- magnitude representation. • EXAMPLE 2.14 • EXAMPLE 2.15 • The signed magnitude has two representations for zero, 10000000 and 00000000 (and mathematically speaking, the simple shouldn’t happen!). 2.4.2 Complement Systems 49 • One’s Complement o This sort of bit-flipping is very simple to implement in computer hardware. o EXAMPLE 2.16 Express 2310 and -910 in 8-bit binary one’s complement form. 2310 = + (000101112) = 000101112 -910 = - (000010012) = 111101102 o EXAMPLE 2.17 o EXAMPLE 2.18 o The primary disadvantage of one’s complement is that we still have two representations for zero: 00000000 and 11111111 • Two’s Complement o Find the one’s complement and add 1. o EXAMPLE 2.19 Express 2310, -2310, and -910 in 8-bit binary two’s complement form. 2310 = + (000101112) = 000101112 -2310 = - (000101112) = 111010002 + 1 = 111010012 -910 = - (000010012) = 111101102 + 1 = 111101112 o EXAMPLE 2.20 o EXAMPLE 2.21 o A Simple Rule for Detecting an Overflow Condition: If the carry in the sign bit equals the carry out of the bit, no overflow has occurred. If the carry into the sign CMPS375 Class Notes Page 4/ 16 by Kuo-pao Yang bit is different from the carry out of the sign bit, over (and thus an error) has occurred. o EXAMPLE 2.22 Find the sum of 12610 and 810 in binary using two’s complement arithmetic. A one is carried into the leftmost bit, but a zero is carried out. Because these carries are not equal, an overflow has occurred. o N bits can represent –(2n-1) to 2n-1 -1. With signed-magnitude number, for example, 4 bits allow us to represent the value -7 through +7. However using two’s complement, we can represent the value -8 through +7. • Integer Multiplication and Division o For each digit in the multiplier, the multiplicand is “shifted” one bit to the left. When the multiplier is 1, the “shifted” multiplicand is added to a running sum of partial products. o EXAMPLE Find the product of 000001102 and 000010112. o When the divisor is much smaller than the dividend, we get a condition known as divide underflow, which the computer sees as the equivalent of division by zero. o Computer makes a distinction between integer division and floating-point division. With integer division, the answer comes in two parts: a quotient and a remainder. Floating-point division results in a number that is expressed as a binary fraction. Floating-point calculations are carried out in dedicated circuits call floating- point units, or FPU. CMPS375 Class Notes Page 5/ 16 by Kuo-pao Yang 2.5 Floating-Point Representation 55 • In scientific notion, numbers are expressed in two parts: a fractional part call a mantissa, and an exponential part that indicates the power of ten to which the mantissa should be raised to obtain the value we need. 2.5.1 A Simple Model 56 • In digital computers, floating-point number consist of three parts: a sign bit, an exponent part (representing the exponent on a power of 2), and a fractional part called a significand (which is a fancy word for a mantissa). 1 bit 5 bits 8 bits Sign bit Exponent Significand FIGURE 2.2 Floating-Point Representation • Unbiased Exponent 5 0 00101 10001000 1710 = 0.100012 * 2 17 0 10001 10000000 6553610 = 0.12 * 2 • Biased Exponent: We select 16 because it is midway between 0 and 31 (our exponent has 5 bits, thus allowing for 25 or 32 values). Any number larger than 16 in the exponent field will represent a positive value. Value less than 16 will indicate negative values. 5 0 10101 10001000 1710 = 0.100012 * 2 The biased exponent is 16 + 5 = 21 -1 0 01111 10000000 0.2510 = 0.12 * 2 • EXAMPLE 2.23 • A normalized form is used for storing a floating-point number in memory. A normalized form is a floating-point representation where the leftmost bit of the significand will always be a 1. Example: Internal representation of (10.25)10 (10.25)10 = (1010.01) 2 (Un-normalized form) 0 = (1010.01) 2 x 2 . 1 = (101.001) 2 x 2 . : 4 = (.101001) 2 x 2 (Normalized form) 5 = (.0101001) 2 x 2 (Un-normalized form) 6 = (.00101001) 2 x 2 .